Question

Find the area of the region that lies inside the first curve and outside the second curve.
r = 1 + cos(θ), r = 2 − cos(θ)

Answer 1

Unless you provide some upper and lower limits, infinite such area exists, and the area will tend to infinity.

Answer 2

First plot the curves. They intersect at
(3/2,π/3) and (3/2,-π/3)
So, let's just take twice the area for θ in [0,π/3]

Now just plug in the functions. The area is

∫[0,π/3] (1+cosθ)^2 - (2-cosθ)^2 dθ
= ∫[0,π/3] 6cosθ-3 dθ = 3√3 - π